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Lau Chi Hin  

The Chinese University of Hong Kong

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1. Sequential and combinatorial games

2. Two-person zero sum games

3. Linear programming and matrix games

4. Non-zero sum games

5. Cooperative games

MATH4250 Game Theory

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Prisoner’s Dilemma 

• John and Peter have been arrested for

possession of guns. The police suspects thatthey are going to commit a major crime.

• If no one confesses, they will both be jailedfor 1 years.

• If only one confesses, he’ll go free and hispartner will be jailed for 5 years.

• If they both confess, they both get 3 year.

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Prisoner’s Dilemma 

Peter

Confess Don’tconfess

JohnConfess 3,3 0,5

Don’t

confess5,0 1,1

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Prisoner’s Dilemma 

• If Peter confesses： 

John “confess” (3 years) better than

“don’t confess” (5 years).• If Peter doesn’t confess： 

John “confess” (0 years) better than

“don’t confess” (1 years).

1,15,0Don’ t

0,53,3Confess

John

Don’ t ConfessPeter

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Prisoner’s Dilemma 

• Thus John should confess whatever Peter does.

•Similarly, Peter should also confess.

1,15,0Don’ t0,53,3ConfessJohn

Don’ t ConfessPeter

Conclusion: Both of them should confess

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Prisoner’s Dilemma 

Peter

Confess Don’tconfess

JohnConfess 3,3 0,5

Don’t

confess5,0 1,1

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 Applications

•

Economics• Political science

• Biology

• Computer science• Philosophy

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 Vickrey Auction

The highest bidder wins, but the

price paid is the second-highest bid.

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 Vickrey Auction

明 報 2009年10月28日 

再論以博弈論打破勾地困局

 政府可考慮 如勾地者最終成功投得地皮 可讓他們享有

3

至

5

％的折扣優惠  如此建議獲接納 發展商會甘心做

「出頭鳥」 搶先以高價勾地。

 

…其他發展商 如出價不及勾出地皮的發展商 已考慮了

市場情況和財政計算 他們亦知其中一個對手享有折扣優

惠 所以要打敗對手 出價只有更進取。…

 

也可考慮將最終成交價訂為拍賣地皮的第二最高出價。」 

撰文

:

陸振球

 

明報地產版主管

)

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Nobel Laureates Related

to Game Theory

•

1994: Nash, Harsanyi, Selten• 1996: Vickrey

• 2005: Aumann, Schelling

• 2007: Hurwicz, Maskin, Myerson• 2012: Shapley, Roth

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 vs

Two supermarkets PN and WC 

are engaging in a price war.

Price War

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• Each supermarket can

choose: high price orlow price.

•

If both choose highprice, then each willearn $4 (million).

Price War

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• If both choose low price, then

each will earn $2 (million).• If they choose different strategies,

then the supermarket choosing

high price will earn $0 (million),while the one choosing low pricewill earn $5 (million).

Price War

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WC

Low High

PNLow 2,2 5,0

High 0,5 4,4

Price War

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WC

Low High

PNLow 2,2 5,0

High 0,5 4,4

Price War

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WC

Low High

PNLow 2,2 5,0

High 0,5 4,4

Price War vs Prisoner Dilemma

Peter

ConfessDon’t

confess

JohnConfess 3,3 0,5Don’t

confess 5,0 1,1

These are called

dominant strategy equilibrium.

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 A Beautiful Mind

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John Nash

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John Nash

• Born in 1928

• Earned a PhD fromPrinceton at 22

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• Late 1950s, Nash left

MIT because of

mental illness.

• It is a miracle that he

can recover twenty

years later.

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• In his 27 pages Ph.D thesis “Non-cooperative

Games”, Nash made very importantcontribution in establishing the mathematical

 principles of  Game Theory.

• In this thesis, Nash greatly extended the work

of John von Neumann whose is the founder

of Game Theory. 

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• In 1994, Nash shared

the Nobel Prize inEconomics with John

C. Harsanyi and

Reinhard Selten

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Nash embedding theorem

Any closed Riemannian n -manifold has a C 1 isometric

imbedding in R 2n .

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 John Nash (Annals of math 1957)Theorem: Every finite n-player

non-cooperative game has a mixed

 Nash equilibrium.

Nash’s Theorem 

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A mixed Nash equilibrium is bothplayers use mixed strategy (1/3,1/3,1/3),

that means having a probability of 1/3

of using each of the three gestures.

Rock-paper-scissors

Column Player

Rock Paper Scissor

Row

Player

Rock (0,0) (-1,1) (1,-1)Paper (1,-1) (0,0) (-1,1)

Scissor (-1,1) (1,-1) (0,0)

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Mixed Nash equilibrium:Row player: (2/3,1/3)

Column player: (2/3,1/3)

Modified rock-paper-scissors

Column Player

Rock ScissorRow

Player

Rock (0,0) (1,-1)

Paper (1,-1) (-1,1)

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Brouwerfixed-point

theorem

Nash’s Proof  

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Brouwer’s fixed-point theorem

Fixed-point theorem: 

Any continuous function

from the n -dimensional unit

ball to itself has at least onefixed-point.

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Consequence of fixed-point theorem

- Everybody

has at leastone bald spot.

- There is at

least one place

on earth with

no wind.

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1. Solutions of combinatorial games

2. Zermelo’s theorem

3. Examples: Take-a-way games, Nim, Hex

Combinatorial games

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A sequential game is agame where one player

chooses his action beforethe others choose their.

Sequential Games

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Sequential Games

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In any finite sequential game with

 perfect information, at least one of the players has a drawing strategy. In

 particular if the game cannot end with

a draw, then exactly one of the players has a winning strategy.

Zermelo’s Theorem 

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Nim

What is the winning strategy of Nim? 

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Hex

Can Hex end with a draw? 

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1. Saddle points

2. Mixed strategies

3. Equilibrium pairs4. Values of games

Two-person zero sum game

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1. Linear programming

2. Duality

3. Minimax theorem

Matrix games

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1. Non-cooperative games

2. Mixed Nash equilibrium

Non-zero sum games

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1. Nash bargaining solution

2. Coalitions

3. Imputations

4. Core

5. Shapley values

Cooperative games

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1. Five players put certain amount of

money from $0 to $1,000 to a pool.

2. The total amount of money in thepool will be multiplied by 3.

3. The money in the pool is thendistributed evenly to the players.

Money Sharing Game

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  This explains why every country

is blaming others instead ofputting more resources toenvironment protection.

Game Theory and Environment

Ideal Situation Nash Equilibrium

Strategy $1,000 $0Payoff $2,000 $0

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一蚊雞或無廣告世界盃 

【明報

18/4/2010】無綫、亞視在轉播世界

盃的處理上與有線再次談不攏。 

有線要求兩家免費台一元的版權費  但就要

把有線世界盃賽事連廣告一齊播  …這等於

讓有線同時出賣無綫、亞視的廣告時間告時

間送給有線。… 有線當然可以把廣告費大大

提高。兩台當然不會應承 有線則可以說兩

台不顧廣大觀眾利益  因這做法對觀眾有利

對有線更有利 只損害兩台收益。

 

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無綫、亞視提出反建議 有線只需提供四場世

界盃的主要賽事給兩台 而兩台則不會在這賽

事中放任何廣告  即不利用世界盃來搵錢 只

求讓更多觀眾可以收看。有線很快便拒絕了兩

台這反建議。

 

筆者認為兩台可播世界盃的可能性愈來愈低 

好看的反而是有線跟兩台互相過招 大家表面

上都以觀眾利益作大前提 內裏當然是希望取

得最大利益。到目前為止 雖然任何方案都是

想更多人看到世界盃 卻沒一個可為雙方接受 

問題當然不在觀眾利益之上。

 

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World Cup Broadcast

Additional payoff  

additional commercial income

Pay TV proposal 

- Put their commercial at Free TV 

- Gain all additional income

Free TV proposal

- Do not put any commercial 

- Abandon all additional income

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三台達協議播放世界盃

 

【明報

 27/4/2010

二

 )

】有線電視終與兩間免費電視

台 就轉播

4

場主要賽事達成協議  無線及亞視將於

數碼頻道播放由有線提供的

4

場直播賽事連廣告。…

 

三個電視台昨日傍晚突然發表聲明  指「基於公眾

利益」達成播放本屆世界盃賽事協議 … 一致感謝

政府居中協助及斡旋。 

有線曾去信兩台 提出只收取象徵式

10

元的轉播費用 

但兩台必須播放有線的世界盃節目 包括廣告。兩

台指有線的建議佔用的廣告時段 故不同意播廣告

如今由數碼頻道播放可算「各退一步」。

 

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NBA談判徹底破裂 (體育)

2011-11-15歷時兩年半的

NBA勞資協議談

判遭遇重挫。球員工會拒絕資方提交的最新修

訂提案 準備解散工會  以《反壟斷法》向資

方提出訴訟。而

NBA主席史坦就警告 如果工

會不接受建議 資方的立場會轉趨強硬。

 

鑑於解散工會和動用法律手段解決勞資糾紛需

要至少數個月 球員的決定很可能意味著

2011至

2012賽季整體報廢。如果真的如此

那將是NBA史上首次因停賽而斷送整個賽季。 

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NBA negotiation

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美國NBA球季有望聖誕重開 

2011-11-27美國NBA勞資談判出

現曙光 勞資雙方經過最近一輪 5

小時的漫長談判 達成框架協議 

常規賽有望在 2月25日開始 但場

數會由82場 縮減至66場

NBA negotiation

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Lloyd Stowell Shapley

• Born in 1923

•His father HarlowShapley is knownfor determining the

position of the Sunin the Milky WayGalaxy

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Lloyd Stowell Shapley

• Drafted when he

was a student atHarvard in 1947

• Served in the Army in Chengdu,

China and received the Bronze Stardecoration for breaking the Japaneseweather code

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• A value for n -person Games (1953)

• College Admissions and the Stability of

Marriage (with Davis Gale 1962)• Awarded Nobel

Memorial Prize

in EconomicSciences withAlvin ElliotRoth in 2012

Shapley Roth

Nobel Prize in Economic 2012

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  This year's Prize concerns a central economic

problem: how to match di fferent agents as well as

possible. For example, students have to be matchedwith schools, and donors of human organs with

patients in need of a transplant. How can such

matching be accomplished as eff iciently as possible?

What methods are beneficial to what groups? Theprize rewards two scholars who have answered these

questions on a journey from abstract theory on stable

allocations to practical design of market insti tutions. 

Nobel Prize in Economic 2012

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  A set of marriages is unstable if

there are two men M  and m  whoare married to two women W  and w , respectively, although W  

prefers m to M and m  prefers W  to w . A set of marriages is stableif it is not unstable.

Stable marriage problem

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Unstable set of marriages

M W m w

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Unstable marriage

M w

m W

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Existence of stable set

Shapley’s Theorem: 

Suppose there are n  men and n  women. There always exists astable set of marriages.

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Ranking matrix

W1 W2 W3

M1 1,3 2,2 3,1

M2 3,1 1,3 2,2

M3 2,2 3,1 1,3

• {(M1,W1), (M2,W2), (M3,W3)} is stable.

(All men with their first choices.)• {(M1,W3), (M2,W1), (M3,W2)} is stable.

(All women with their first choices.)

• {(M1,W1), (M2,W3), (M3,W2)} is unstable.(Consider (M3,W1).)

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Deferred-acceptance procedure

W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Alternation of

• Men propose to their favorite women.

• Women reject unfavorable men.

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Deferred-acceptance procedure

Step 1: Men propose to their favorite women.

(M1,W1),(M2,W2),(M3,W4),(M4,W1)

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Step 2: Women reject unfavorable men.

(M1,W1),(M2,W2),(M3,W4),(M4,W1)

Deferred-acceptance procedure

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Step 3: Men propose to their favorite women.

(M1,W1),(M2,W2),(M3,W4),(M4,W4)

Deferred-acceptance procedure

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Step 4: Women reject unfavorable men.

(M1,W1),(M2,W2),(M3,W4),(M4,W4)

Deferred-acceptance procedure

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Step 5: Men propose to their favorite women.

(M1,W1),(M2,W2),(M3,W1),(M4,W4)

Deferred-acceptance procedure

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Step 6: Women reject unfavorable men.

(M1,W1),(M2,W2),(M3,W1),(M4,W4)

Deferred-acceptance procedure

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Step 7: Men propose to their favorable women.

(M1,W2),(M2,W2),(M3,W1),(M4,W4)

Deferred-acceptance procedure

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Step 8: Women reject unfavorable men.

(M1,W2),(M2,W2),(M3,W1),(M4,W4)

Deferred-acceptance procedure

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Step 9: Men propose to their favorite women.

(M1,W2),(M2,W1),(M3,W1),(M4,W4)

Deferred-acceptance procedure

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Step 10: Women reject unfavorable men.

(M1,W2),(M2,W2),(M3,W1),(M4,W4)

Deferred-acceptance procedure

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

Step 11: Men propose to their favorite women.

(M1,W2),(M2,W3),(M3,W1),(M4,W4)

Deferred-acceptance procedure

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W1 W2 W3 W4

M1 1,2 2,1 3,2 4,1

M2 2,4 1,2 3,1 4,2

M3 2,1 3,3 4,3 1,4

M4 1,3 4,4 3,4 2,3

A stable set of marriages is

(M1,W2),(M2,W3),(M3,W1),(M4,W4)

Note: This example has only one stable set.

Deferred-acceptance procedure

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

A stable set of stable marriages is(M1,W1),(M2,W3),(M3,W2),(M4,W4)

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

Of course, we may ask the women to propose first.

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

Then the men reject their unfavorable women.

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

We obtain another stable set of marriages(M1,W1),(M2,W2),(M3,W4),(M4,W3)

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 Another example

W1 W2 W3 W4

M1 3,1 1,3 4,1 2,4

M2 1,4 3,1 2,4 4,1

M3 4,2 1,2 2,3 3,2

M4 3,3 1,4 4,2 2,3

We see that stable set of marriages is not unique(M1,W1),(M2,W2),(M3,W4),(M4,W3)

(M1,W1),(M2,W3),(M3,W2),(M4,W4)

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B1 B2 B3 B4

B1 1,2 2,1 3,1

B2 2,1 1,2 3,2

B3 1,2 2,1 3,3

B4 1,3 2,3 3,3

Problem of roommates

An even number of boys are divided up intopairs of roommates.

The boy pairs with B4 will have a better option.Stable set of pairing does not always exist.

S l i f

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Solution of games

• Non-cooperative game:

Mixed Nash equilibrium• Non-transferable utility cooperative game:

Nash bargaining solution

•

Transferable utility cooperative game:Core, Stable set, Shapley value,… 

h l l

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Shapley value

The Shapley value of player k  is defined as

S k 

n

S nS 

 N S k 

  ,!

!!1

   

where

Shapley value of player k  indicates the average

contribution of player k  to all orders of coalitions. 

  }){\()(,   k S vS vS k     

is the contribution of player k  to the coalition S .

2

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2-person cooperative game

For 2-person games, the playersshare evenly the additional payoffgained by cooperation.

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Restaurant Coupon

Suppose Rose has a coupon

 R A  I N BOW CAFÉ 20% off for single50% off for couple

Rose invited Colin to dinner at Rainbow

café. They plan to spend $100 each before

discount. How should they split the bill?

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Restaurant Coupon

Coalition Original Need to pay v (S )

{R} 100 80 20

{C} 100 100 0{R,C} 200 100 100

C

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Restaurant Coupon

402

201000

60

2

2010020

C 

 R

 

 

Rose should pay $40 and Colin

should pay $60.

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Sharing taxi fare

Andy, Betty and Cindy, want to go to City One, TaiWai and Tsuen Wan respectively from CUHK by

taxi. The taxi fares are given in the following table.

Destination Fare

City One $50

Tai Wai $60

Tsuen Wan $120

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Sharing taxi fare

CUHK 

 

City One

Tai WaiTsuen Wan

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Sharing taxi fare

However, they can save some money by hiringa taxi together and sharing the taxi fare.

Destination (S ) Fare Save (v (S ))

City One & Tai Wan $80 $50+$60-$80=$30

City One & Tsuen Wan $150 $50+$120-$150=$20

Tai Wan & Tsuen Wan $130 $60+$120-$130=$50

All 3 places $160 $50+$60+$120-$160=$70

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Sharing taxi fare

Player’s contribution to orders of coalitions 

Order Player 1 (Andy) contributionABC 0 

ACB 0 

BAC 30

BCA 70-50=20

CAB 20

CBA 70-50=20

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  The additional payoff of Andy is

Andy should pay $50 - $15 = $35

15

6

2020302000

1

 

Sharing taxi fare

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Destination Original Save PayAndy City One $50 $15 $35

Betty Tai Wai $60 $30 $30

Cindy Tsuen Wan $120 $25 $95

Sharing taxi fare